Twin Paradox Problem: Do Twins Age Differently?

[TrickyDicky]

When we're talking about the SR solution, either we're conducting a thought experiment in an idealized flat space or we're conducting a real experiment in an environment where the gravitational effects are negligible. Either way, we use the Minkowski metric. (That's pretty much the definition of special relativity as a special case of the general theory).

We calculate the time elapsed for each twin as \tau = \int_A^B \sqrt{g_{ij} dx^i dx^j} where the g_ij are the Minkowski metric components... and that calculation requires no actual inertial reference.

[Thanks to stevndaryl for the plagiarized latex]

Of course you can substitute the inertial reference in SR with Minkowski metric tensor, that is known since 1907.

 

[harrylin]

For me, "objective" in the context of SR is similar to "absolute": if all inertial frames agree that a statement is true. [..]

I don't consider that good enough for a statement to be objective. I think that it must also be the case that the terms mentioned in that statement have a meaning that is independent of observers. [..]

Perhaps for you an "objective statement" may only relate to invariants? I'm not that demanding... Check,http://dictionary.reference.com/browse/objective?s=t :
5. not influenced by personal feelings, interpretations, or prejudice; based on facts; unbiased

Based on the dictionary, I conclude that definitely also the sense in which robinpike seems to have meant the statement of post #68, is "objective". It's merely a different kind of "being objective" than yours.

Even more, with your definition, SR-type "time dilation" isn't even possibly part of an "objective statement", or am I mistaken?

 

[ghwellsjr]

ghwell, You completely missed my point, see my previous posts.
I was referring to the fact that there must be some inertial frame in the problem, not that it must be one of the twins.

I can only go by what you say, not by what you are thinking. But even the statement that there must be some inertial frame in the problem is not correct. There doesn't have to be any frame. Consider this:

Two observers are traveling toward each other at a relative speed of 90%c. They each observe the other ones clock running at 4.359 times their own. When they pass, they reset their clocks to zero. Now they each observe the other ones clock running at 0.2294 times their own. After a while, one of them turns around such that they are now approaching each other at 90%c and, like initially, the one that turned around immediatly sees the other ones clock running at 4.359 times his own. When they pass again, they compare the accumulated times on their respective clocks. The one that turned around sees the time on the other ones clock as 2.2942 times his own. You will note that this is exactly a restatement of Rishavutkarsh's problem in which he did not specify an inertial frame and you will note that I did not use any frame in my analysis in post #5, inertial or non-inertial.

If you want you can use any frame to analyze the problem that Rishavutkarsh stated but it will not provide any more insight or information into what is happening. For example, you could use a frame in which the inertial observer remains at rest. Or you can use a frame in which the non-inertial observer is at rest during the first part of the scenario. Or you can use a frame in which the non-inertial observer is at rest during the last part of the scenario. Or you can use a frame in which during the first part of the problem both observers are traveling in opposite directions at the same speed or in which this is true for the last part of the problem. And note that I always say "a" frame for each of these case because there are an infinite number of frames for each one that you can choose from. Frames are arbitrary and they don't change or influence what the observers see, measure, or observe. No frame is preferred over any other, even the rest frame(s) of the observers.

 

[ghwellsjr]

What you need to compute proper times is a coordinate system with a known metric tensor.

No you don't, I just did it in post #5 and again in post #93.

 

[stevendaryl]

Perhaps for you an "objective statement" may only relate to invariants?

I consider other things besides invariants to be objective, namely vectors and tensors. They have a meaning that is independent of observer (although their components are relative to a coordinate system).

I'm not that demanding... Check,http://dictionary.reference.com/browse/objective?s=t :
5. not influenced by personal feelings, interpretations, or prejudice; based on facts; unbiased

Well, it seems to me that the term "clock rate" is a subjective notion, and so IS based on interpretations.

Based on the dictionary, I conclude that definitely also the sense in which robinpike seems to have meant the statement of post #68, is "objective". It's merely a different kind of "being objective" than yours.

Well, it seems to me that you can always paraphrase a statement that is objective in your sense into a statement that is objective in my sense. You can say "For any coordinate system, there is a time at which the clock rate of the traveling twin is less than the clock rate of the stay-at-home twin, according to that coordinate system." That's objectively true.

Even more, with your definition, SR-type "time dilation" isn't even possibly part of an "objective statement", or am I mistaken?

There is a corresponding objective statement, which is that an inertial path connecting two spacetime events has a greater proper time than an accelerated path connecting the same two points.

 

[stevendaryl]

I wrote:

What you need to compute proper times is a coordinate system with a known metric tensor.

No you don't, I just did it in post #5 and again in post #93.

No, you didn't. You wrote:

Two observers are traveling toward each other at a relative speed of 90%c. They each observe the other ones clock running at 4.359 times their own. When they pass, they reset their clocks to zero. Now they each observe the other ones clock running at 0.2294 times their own.

How did you compute those two numbers, 4.359 and 0.2294, if not by using a metric?

Maybe you are just saying that you can observe the effects of time dilation without using any metric. I wouldn't say that you were computing it.

Maybe I should rephrase what I was saying. If you want to PREDICT how much elapsed time will occur on a moving clock, then you need to know a coordinate system for describing the clock's motion, and you need to know a metric for that coordinate system.

 

[TrickyDicky]

...even the statement that there must be some inertial frame in the problem is not correct. There doesn't have to be any frame. you will note that I did not use any frame in my analysis in post #5, inertial or non-inertial.
...

No frame is preferred over any other, even the rest frame(s) of the observers.

These statements don't make any sense within SR.
As soon as you are talking about percentages of c you are using inertial frames.
The last statement is true as long as there are only inertial frames, all equally valid of course.

 

[D H]

How did you compute those two numbers, 4.359 and 0.2294, if not by using a metric?

No metric is needed. For example, suppose the two ships regularly send time tagged messages to one another. Each message is time tagged with a time of transmission by the sending ship per that ship's clock. A time of receipt is added to the message by the receiving ship per that ship's clock. Computing those rates is simple; it's just a matter of looking over the message log files and comparing transmission times versus reception times. When the ships are approaching one another, each ship will say the other ship's clock is running fast; when moving away from one another, each ship will say the other ship's clock is running slow.

 

[ghwellsjr]

These statements don't make any sense within SR.
As soon as you are talking about percentages of c you are using inertial frames.
The last statement is true as long as there are only inertial frames, all equally valid of course.

Yes, within SR, you need to use frames. But you don't need to use SR to present or solve every relativistic problem which was the case in this thread. Of course, you can use SR if you want.

I disagree that talking about percentages of c means that I am using either SR or frames. Are you saying that prior to Einstein, nobody could talk about percentages of c? Never heard of such a thing. And if what you say is true, then what frame was I talking about in my response to you in post #93?

I wish your point that SR permits only inertial frames so that we could assume an inertial frame whenever in the context of SR we are talking about frames but unfortunately, I have learned that non-inertial frames are also just as valid within the context of SR.

 

[stevendaryl]

No metric is needed. For example, suppose the two ships regularly send time tagged messages to one another. Each message is time tagged with a time of transmission by the sending ship per that ship's clock. A time of receipt is added to the message by the receiving ship per that ship's clock. Computing those rates is simple; it's just a matter of looking over the message log files and comparing transmission times versus reception times. When the ships are approaching one another, each ship will say the other ship's clock is running fast; when moving away from one another, each ship will say the other ship's clock is running slow.

Yes, if you have a detailed record of transmission and reception times on the two ships, you can use that to demonstrate time dilation. What I mean is that there is no way to compute those transmission and reception times without using a coordinate system and a metric for that coordinate system. If I just tell you that I have two ships that start together, move apart, and then come back together, can you tell me what the elapsed times on the two ships will be? No, not without more information.

 

[TrickyDicky]

Yes, within SR, you need to use frames. But you don't need to use SR to present or solve every relativistic problem which was the case in this thread. Of course, you can use SR if you want.

I disagree that talking about percentages of c means that I am using either SR or frames. Are you saying that prior to Einstein, nobody could talk about percentages of c? Never heard of such a thing. And if what you say is true, then what frame was I talking about in my response to you in post #93?

I wish your point that SR permits only inertial frames so that we could assume an inertial frame whenever in the context of SR we are talking about frames but unfortunately, I have learned that non-inertial frames are also just as valid within the context of SR.

Classical Newtonian mechanics assumed infinite light-speed, it wasn't until Maxwell eq. that a finite constant c came to dispute that to Newton, and precisely that discrepancy triggered Einstein SR. Empirically of course it was known before Maxwell and Einstein that light speed was not infinite, but had not been introduced yet in the scientific equations of mechanics.

I never said that SR only admits inertial frames but that when other frames are used, inertial ones are the preferred ones.

 

[TrickyDicky]

Yes, if you have a detailed record of transmission and reception times on the two ships, you can use that to demonstrate time dilation. What I mean is that there is no way to compute those transmission and reception times without using a coordinate system and a metric for that coordinate system. If I just tell you that I have two ships that start together, move apart, and then come back together, can you tell me what the elapsed times on the two ships will be? No, not without more information.

Exactly, and this extra information is surely needed to solve the twin paradox, that is, to break the symmetry.

 

[harrylin]

[..] Well, it seems to me that you can always paraphrase a statement that is objective in your sense into a statement that is objective in my sense. You can say "For any coordinate system, there is a time at which the clock rate of the traveling twin is less than the clock rate of the stay-at-home twin, according to that coordinate system." That's objectively true.

Two people, one idea - I had in mind a similar reformulation of his original statement with which I think everyone would agree.

Reformulated with greater economy of words:

"it can be deduced that according to any inertial coordinate system the clock rate of the traveling twin's clock must have slowed down at some point during the journey."

As a matter of fact, that was the way in which it made "objective sense" to me; I took "any inertial coordinate system" as implied.

 

[harrylin]

Classical Newtonian mechanics assumed infinite light-speed [..] [that speed] had not been introduced yet in the scientific equations of mechanics.

That is definitely wrong: it was included since Huygens and Newton.

Right. [Without any idea of the state of motion of the participating objects, little can be predicted with certainty about the physical effects. In that sense is motion not just "relative"]

As a matter of fact, it was the very purpose of Langevin's "twin" example to demonstrate the "absoluteness" of non-inertial motion; and apparently it was that issue that led to the "paradox". As his paper is a bit long-winded I'll give a synopsis that is focussed on the topic here:

The Foucault pendulum and the gyroscope demonstrate that although uniform translation has no absolute sense, rotation does have such an absolute sense.

For systems in uniform translational motion it is as if they are stationary relative to the aether: uniform motion in the aether has no experimental sense. However, one should not conclude that the aether concept must be abandoned, that the aether is non-existent and inaccessible to experiment. Uniform velocity relative to it cannot be detected, but any change of velocity, any acceleration has an absolute sense. We have therefore hold on the ether by means of accelerations; acceleration has an absolute sense as it causes the production of electromagnetic waves by matter that undergoes a change in velocity, and the ether manifests its reality as a vehicle, as support for the energy that is carried by these waves. We will see this absolute character of acceleration manifest itself in another form.

Whoever of us who is willing could explore the future Earth by making a leap forward in time which for the Earth will last two centuries but for him will last two years, however without any hope of coming back to inform us of the outcome of the voyage.
He should agree to be locked up in a projectile that will be launched from Earth with a velocity close to that of light, arranging an encounter with a star after one year of the traveller's life and which sends him back to Earth with the same velocity. Back on Earth, he will find our world aged by two hundred years if his velocity was only one twenty-thousandth less than the velocity of light. It is fun to describe how our explorer and the Earth would see each other live if they could [etc.,]
- http://en.wikisource.org/wiki/The_Evolution_of_Space_and_Time

This demonstration of the "absoluteness" of acceleration became paradoxical ("Twin Paradox") due to Einstein's contrary claim that "all motion is relative".

Einstein defended that contrary view in 1918 with the following paper:

http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity

The historical Twin paradox is essentially about Newton vs. Mach (or "absolutism" against "relativism"), as presented by Langevin vs. Einstein.

 

[TrickyDicky]

That is definitely wrong: it was included since Huygens and Newton.

Nope, you got that wrong about Newton mechanics and I didn't mention optics or Huygens.

 

[harrylin]

Nope, you got that wrong about Newton mechanics and I didn't mention optics or Huygens.

Newton thought that light consisted of particles that travel at limited speed in vacuum, and faster in glass. Perhaps you did not mean equations of mechanics but system transformation equations? Those did not include light speed at all and had nothing to do with it.

 

[TrickyDicky]

I said nothing about what Newton thought, and explicitly said that of course people knew light had finite speed long before.
Are transformation equations not equations of mechanics?
Do you mean you have never heard about the difference between Galilei relativity in classical mechanics and the special relativity of Einstein?

 

[harrylin]

I said nothing about what Newton thought, and explicitly said that of course people knew light had finite speed long before.
Are transformation equations not equations of mechanics?
Do you mean you have never heard about the difference between Galilei relativity in classical mechanics and the special relativity of Einstein?

Sure I did, and I also know very well some of the nonsense that is said about it. Newton did not include the idea of infinite light speed in any of his equations. However, that has nothing to do with the topic and I may have now traced your disagreement with George to another wrong phrasing that has nothing to do with what I think you meant to say. So, to limit the accumulation of waste of time, please tell me if this is correct:

When you wrote:
"As soon as you are talking about percentages of c you are using inertial frames",
did you mean that "0.9c" unwittingly refers to one or two inertial frames? If so, you may be right about that. I don't see how "0.9c" can be determined without referral to a reference system.

 

[TrickyDicky]

Newton did not include the idea of infinite light speed in any of his equations.

And I never said that he included it, but that classical mechanics assumed galileo relativity in which light speed is treated as infinite.

However, that has nothing to do with the topic and I may have now traced your disagreement with George to another wrong phrasing that has nothing to do with what I think you meant to say. So, to limit the accumulation of waste of time, please tell me if this is correct:

When you wrote:
"As soon as you are talking about percentages of c you are using inertial frames",
did you mean that "0.9c" unwittingly refers to one or two inertial frames? If so, you may be right about that. I don't see how "0.9c" can be determined without referral to a reference system.

Yes, something like that.

 

[zonde]

Certainly I was not clear enough in posts #8 and #27, but what was not clear??

I reread you post #8 and I can say that when I read it first time I didn't get that you are talking only about Earth frame.

At the moment that you make a turn-around, you:
1. can not influence what happens on earth
2. have only one inertial reference system at your disposal, which is the one of the Earth (ECI frame).
Next, after the turn-around you can decide to still indirectly use the ECI frame (just as astronauts always have done until now in real life), or set up a new inertial reference system by re-synchronizing your clocks. That system maps a different distant time as the other ones.

When that is understood, it is immediately clear that it's just a matter of switching reference frames, so that alternative scenario's with fly-by at the same velocities cannot have a different effect. There is no problem with that illustration, but it should not be presented as spooky action at a distance.

I said: "This "jump" ahead is a coordinate effect." I suppose this clearly says it's not physical effect.

Anyways, my objection is that you analyze "twin paradox" only from one inertial frame. I don't know what was historical role of that paradox but I suppose that all current discussion around "twin paradox" are concerned about consistency of SR.
Obviously to convince anybody about consistency of SR you have to present two alternative ways (involving different reference frames) how you can get to the same result i.e. you have to analyze "twin paradox" from perspective of both twins in parallel.

It's clear how it looks from perspective of first (stay at home) twin - second twin is time dilated by the same factor in forward and backward trip (the same speed). Acceleration does not come into the picture.

Now how it looks from perspective of second twin - first twin is time dilated by the same factor in forward and backward trip (the same speed). But contrary to the first case acceleration (because we have to switch coordinates) has effect. And result of this clearly unphysical effect is that we just add some accumulated time to Earth clock.

 

[zonde]

This is just another of the many ways to analyze the Twin Paradox and they all agree, as you pointed out, concerning their picture of Earth when they meet. And they all agree with the final outcome. And they all agree with everything else in between that is observable. They don't agree on what you are calling remote "now" which is another way of saying "coordinate time" but that is consistent with the calculation of the Proper Time on both clocks. The coordinate times can vary all over the place between these different frames but when you apply the time dilation you get the same Proper Time at each event no matter what frame you use.

You have proposed three inertial observers. You could have proposed analyzing what happens according to each of their rest frames and there would be no frame jumping and no acceleration. I hope you're not suggesting that these three inertial frames are not all equally valid and I hope you're not suggesting that an analysis based on jumping between two of those frames is somehow more valid or better suited to explaining what is "really" happening in the Twin Paradox, are you?

If you want to get some intuition about LT you would want to do some "jumping" between frames, right?

If the traveling twin actually knows physics, he would be aware that there is no such thing as the "real pace of the Earth twin's clock while traveling towards him". He would know that he can analyze the pace of both of their clocks from any inertial frame of reference and each one can assign different paces to their two clocks, none of which can be considered "real". What's real is the visual data that you call an illusion. Furthermore, each one of these inertial reference frames will agree on exactly what each twin sees throughout the entire trip. You can also analyze the scenario from non-inertial frames or jumping inertial frames and they can assign completely different paces to the two clocks but they will all agree on what each twin really sees.

Do you completely agree with everything I said in the above quote?

I agree with everything except statement in bold.

If you do, then please read this quote from post #42:

You just said that real meant within a given inertial frame of reference and now you want to talk about a frame that the traveling twin is a rest in. But it cannot be an inertial frame for the entire trip so how does that work?

I don't know why you want to make this so complicated. Let's do what you said and pick as our given inertial frame of reference the one in which the Earth twin is at rest and in which the traveling twin starts out and ends up at rest. In this frame the Earth twin's clock runs normally.

Now the traveling twin accelerates instantly to a speed of 90%c. Gamma at this speed is 2.294 (not 0.4359 as you claim in your linked diagram). That means that a clock traveling at 90%c will run slower by a factor of 2.294. The traveling twin's clock will run slower than the Earth twin's clock by this amount during his entire trip so when he gets back the Earth clock has advanced by 2.294 times whatever his clock advanced. This is what really happens according to your definition of real and it's exactly what I said would happen in post #5 and so I don't know why you say it's misleading.

Do you completely agree with everything I said in the above quote?

I will leave this without answer. There is some discussion involving definitions of real and I will try to stay out of this (within this thread).

If you do, then don't you think it is important to point out that whatever frame provides us with the simplest way to determine what will happen is just as valid as any other frame(s) and no other analysis based on any other frame(s) will provide us with any additional insight or information into what is happening or what any observer observes and so there is no point in discussing other frame(s) except to show that they all agree on what each observer observes throughout the entire scenario?

Yes, I agree. This is important. We can pick whatever frame we like for analysis. But if we want to demonstrate consistency of SR (for those who are not sure about this) we can compare any frames using LT.

 

[harrylin]

And I never said that he included it, but that classical mechanics assumed galileo relativity in which light speed is treated as infinite.

Yes, that's what I thought that you said; and I said that the Galilean relativity does not treat light speed at all.

Yes, something like that.

Good - then the discussion is back on track.

 

[TrickyDicky]

Galilean relativity does not treat light speed at all.

This is basic classical mechanics.
In which time is absolute, (a parameter, not a dimension as in Minkowskian SR), and space is Euclidean, this implies that light speed is infinite, there is no relativity of simultaneity, it takes no time for light signals to travel, now is the same now for anybody. Is this so hard to understand? Ask anyone versed on classical mechanics if you don't believe me.

 

[harrylin]

I reread you post #8 and I can say that when I read it first time I didn't get that you are talking only about Earth frame. [..]

The Earth is not an inertial frame, while in SR everything is analyzed with inertial frames; I next tried to clarify in post #27 that any inertial frame and any amount of switching between such frames can be used for the analysis, because that is how such transformations work. And we are free to map ourselves in any of those.

Anyways, my objection is that you analyze "twin paradox" only from one inertial frame. [...]

No, you took that message out of context: it just explains that the Earth doesn't need to be treated like an inertial frame, in response to an issue that TrickyDicky raised (but he next said that he meant a different issue, and that issue relates to the original purpose of the "twin" example). I already stressed in post #8 that one can choose to use any combination of inertial frames that one likes.

The past threads explain how the group property of the Lorentz transformations works by means of more than enough examples (linked to in post #3). Surely nobody wants to go through the motions again, as if it is not sufficiently explained?

[edit: I now removed my last remark which probably just related to how different people phrase the same thing differently]

 

[harrylin]

This is basic classical mechanics.
In which time is absolute, (a parameter, not a dimension as in SR), and space is Euclidean, this implies that light speed is infinite, there is no relativity of simultaneity, it takes no time for light signals to travel, now is the same now for anybody. Is this so hard to understand? Ask anyone versed on classical mechanics if you don't believe me.

I am rather versed in classical mechanics and that's what I referred to in my post [edit: post #108]; if you like we can discuss that in a topic on that issue (we must not hijack this thread; and perhaps this was already discussed in the past, in which case we can just continue an old thread on that topic).

 

[ghwellsjr]

If you want to get some intuition about LT you would want to do some "jumping" between frames, right?

The term "frame jumping" means explaining a scenario using two or more inertial frames where each is used for a different part of the scenario. For example, using one inertial frame for the traveling twin during his outbound portion of the trip and a second inertial frame during the inbound portion of the trip. I'm not aware of how the Lorentz Transformation process would be involved. Can you explain?

...
We can pick whatever frame we like for analysis. But if we want to demonstrate consistency of SR (for those who are not sure about this) we can compare any frames using LT.

Since the Lorentz Transformation process is pure algebra, how does it demonstrate consistency of SR?

EDIT: After posting this response to your response to me in post #111, I see that you provided the answers to my questions in post #110, so you don't need to answer here, I will compose another post in response to #110 (even though it was not directed at me).

 

[TrickyDicky]

that's what I referred to in my post;

What post? And if so, why did you keep saying I was wrong if that was what you referred to in your post?

 

[harrylin]

What post? And if so, why did you keep saying I was wrong if that was what you referred to in your post?

Fixed now. If you start that topic, we can properly discuss it; I won't further hijack this thread with that topic.

 

[ghwellsjr]

Anyways, my objection is that you analyze "twin paradox" only from one inertial frame. I don't know what was historical role of that paradox but I suppose that all current discussion around "twin paradox" are concerned about consistency of SR.

Obviously to convince anybody about consistency of SR you have to present two alternative ways (involving different reference frames) how you can get to the same result i.e. you have to analyze "twin paradox" from perspective of both twins in parallel.It's clear how it looks from perspective of first (stay at home) twin - second twin is time dilated by the same factor in forward and backward trip (the same speed). Acceleration does not come into the picture.

Just because you analyze the Twin Paradox from a frame in which the Earth twin is at rest and all the time dilation applies to the traveling twin, this does not mean that this frame provides the Earth twin with any insight into the traveling twin's time dilation. Time dilation cannot be observed, it can only be calculated based on a reference frame. What can be observed by the Earth twin is the Relativistic Doppler which this frame will allow you to calculate (although there is an easier way). This is the only perspective that the Earth twin has of the traveling twin's clock.

In the same way, this frame will also allow you to calculate what the traveling twin sees of the Earth twin's clock. They both will see the other one's clock going slower than their own by exactly the same amount, and it's not the time dilation factor since it is different for each one in this frame.

Now how it looks from perspective of second twin - first twin is time dilated by the same factor in forward and backward trip (the same speed). But contrary to the first case acceleration (because we have to switch coordinates) has effect. And result of this clearly unphysical effect is that we just add some accumulated time to Earth clock.

Here is where you have chosen to jump frames so that the traveling twin is always at rest. But before doing that, I want to make sure you agree that we can use a frame in which the traveling twin is at rest during the outbound portion of the trip but for the entire trip. Do you agree with that?

In this case, the traveling twin will have no time dilation for the first half of the trip while the Earth twin has the same time dilation that the traveling twin had in the first frame where the Earth twin was at rest, correct? But then when the traveling twin turns around, he will have more time dilation than the Earth twin continues to have, correct? Do you see this as a legitimate explanation? And do you also see that the this frame does not provide either twin with any more perspective or insight or observation than they had with the first frame? And do you understand that even in this single inertial frame, we can calculate exactly what each twin can observe of each others clock during the entire trip and it will be exactly the same as what we calculated in the first frame?

Now we can go on to a third frame in which the traveling twin is at rest during the inbound portion of the trip but we will apply it during the entire trip. And all the same sorts of questions and answers apply, correct?

Finally, since you want to have the traveling twin and the Earth twin jump frames at the moment of turn around, I beg you to provide us with the details of the calculations. Let's assume that the traveling twin turns around after one year on his clock and is traveling at 90%c. Can you do that? And can you also show the calculations for what each twin sees of the other twin's clock during the entire scenario, please?

And then, to address your comments to me, I'd like you to show us how you use the LT in this process, OK?

 

[TrickyDicky]

Fixed now. If you start that topic, we can properly discuss it; I won't further hijack this thread with that topic.

Admitting your mistake doesn't hijack any thread. When you decided to answer something I wrote in response to ghwellsjr, not you, you must have not thought you were hijacking anything, curiously it only concerns you when it is clear what I said was right.